joint cumulative distribution function

Let $X_{1},X_{2},...,X_{n}$ be $n$ random variables all defined on the same probability space. The joint cumulative distribution function of $X_{1},X_{2},...,X_{n}$, denoted by $F_{X_{1},X_{2},...,X_{n}}(x_{1},x_{2},...,x_{n})$, is the following function:

$F_{X_{1},X_{2},...,X_{n}}:R^{n}\to R$
$F_{X_{1},X_{2},...,X_{n}}(x_{1},x_{2},...,x_{n})=P[X_{1}\leq x_{1},X_{2}\leq x% _{2},...,X_{n}\leq x_{n}]$

As in the unidimensional case, this function satisfies:

1. 1.

$\lim_{(x_{1},...,x_{n})\to(-\infty,...,-\infty)}{F_{X_{1},X_{2},...,X_{n}}(x_{% 1},...,x_{n})}=0$ and $\lim_{(x_{1},...,x_{n})\to(\infty,...,\infty)}{F_{X_{1},X_{2},...,X_{n}}(x_{1}% ,...,x_{n})}=1$

2. 2.

$F_{X_{1},X_{2},...,X_{n}}(x_{1},...,x_{n})$ is a monotone, nondecreasing function.

3. 3.

$F_{X_{1},X_{2},...,X_{n}}(x_{1},...,x_{n})$ is continuous from the right in each variable.

The way to evaluate $F_{X_{1},X_{2},...,X_{n}}(x_{1},...,x_{n})$ is the following:

$F_{X_{1},X_{2},...,X_{n}}(x_{1},...,x_{n})=\int_{-\infty}^{x_{1}}{\int_{-% \infty}^{x_{2}}{\cdots\int_{-\infty}^{x_{n}}{f_{X_{1},X_{2},...,X_{n}}(u_{1},.% ..,u_{n})du_{1}du_{2}\cdots du_{n}}}}$

(if $F$ is continuous) or

$F_{X_{1},X_{2},...,X_{n}}(x_{1},...,x_{n})=\sum_{i_{1}\leq x_{1},...,i_{n}\leq x% _{n}}{f_{X_{1},X_{2},...,X_{n}}(i_{1},...,i_{n})}$

(if $F$ is discrete),

where $f_{X_{1},X_{2},...,X_{n}}$ is the joint density function of $X_{1},...,X_{n}$.

Title joint cumulative distribution function JointCumulativeDistributionFunction 2013-03-22 11:54:52 2013-03-22 11:54:52 mathcam (2727) mathcam (2727) 9 mathcam (2727) Definition msc 60A10 joint cumulative distribution